Problem: Add the following rational expressions. $\dfrac{-6y^2}{8y+5}+\dfrac{3}{y^4}=$
We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({8y+5})\cdot({y^4})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{-6y^2}{{8y+5}}+\dfrac{3}{{y^4}} \\\\ &=\dfrac{-6y^2\cdot({y^4})}{({8y+5})\cdot({y^4})}+\dfrac{3\cdot({8y+5})}{({y^4})\cdot({8y+5})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{-6y^2\cdot(y^4)}{(8y+5)\cdot(y^4)}+\dfrac{3\cdot(8y+5)}{(y^4)\cdot(8y+5)} \\\\ &=\dfrac{-6y^2\cdot(y^4)+3\cdot(8y+5)}{(8y+5)(y^4)} \\\\ &=\dfrac{-6y^6+24y+15}{(8y+5)(y^4)} \end{aligned}$ In conclusion, $\dfrac{-6y^2}{8y+5}+\dfrac{3}{y^4}=\dfrac{-6y^6+24y+15}{(8y+5)(y^4)}$